Editing Spinor (section)

====Representation theoretic point of view====
From a [[representation theory|representation theoretic]] point of view, one knows beforehand that there are some representations of the [[Lie algebra]] of the [[orthogonal group]] that cannot be formed by the usual tensor constructions. These missing representations are then labeled the '''[[spin representation]]s''', and their constituents ''spinors''. From this view, a spinor must belong to a [[group representation|representation]] of the [[covering space|double cover]] of the [[special orthogonal group|rotation group]] {{math|SO(''n'',<math>\Reals</math>)}}, or more generally of a double cover of the [[generalized special orthogonal group]] {{math|SO<sup>+</sup>(''p'', ''q'', <math>\Reals</math>)}} on spaces with a [[metric signature]] of {{math|(''p'', ''q'')}}. These double covers are [[Lie groups]], called the [[spin group]]s {{math|Spin(''n'')}} or {{math|Spin(''p'', ''q'')}}. All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued [[projective representation]]s of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.)

In summary, given a representation specified by the data <math>(V,\text{Spin}(p,q), \rho)</math> where <math>V</math> is a vector space over <math>K = \mathbb{R}</math> or <math>\mathbb{C}</math> and <math>\rho</math> is a homomorphism <math>\rho:\text{Spin}(p,q)\rightarrow \text{GL}(V)</math>, a '''spinor''' is an element of the vector space <math>V</math>.